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Rationality of extended unipotent characters

Olivier Dudas, Gunter Malle

Abstract

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof we use realisations of characters in $\ell$-adic cohomology groups of Deligne--Lusztig varieties as well as block theoretic considerations.

Rationality of extended unipotent characters

Abstract

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof we use realisations of characters in -adic cohomology groups of Deligne--Lusztig varieties as well as block theoretic considerations.
Paper Structure (8 sections, 12 theorems, 6 equations)

This paper contains 8 sections, 12 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Cuspidal unipotent characters
  3. Deligne--Lusztig varieties and unipotent characters
  4. Reduction to simple groups
  5. Eigenvalues of $F$ and character extensions
  6. Extensions of cuspidal unipotent characters
  7. Harish-Chandra theory
  8. Exceptional graph automorphisms

Key Result

Theorem 1

Let ${\mathbf{G}}$ be a simple algebraic group with a Frobenius map $F$ and a commuting non-trivial graph automorphism $\sigma$. Then any cuspidal unipotent character $\rho$ of $G={\mathbf{G}}^F$ has an extension ${\widehat{\rho}}$ to $G\langle\sigma\rangle$ with ${\mathbb{Q}}({\widehat{\rho}})={\ma

Theorems & Definitions (29)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 19 more